Authors
 Shahriar Osfouri ^{} ^{1}
 Reza Azin ^{2}
^{1} Department of Chemical Engineering, Faculty of Petroleum, Gas and Petrochemical Engineering, Persian Gulf University, 7516913897 Bushehr, Iran
^{2} Department of Petroleum Engineering, Faculty of Petroleum, Gas and Petrochemical Engineering, Persian Gulf University, 7516913897 Bushehr, Iran
Abstract
Characterization of heavy end, as plus fraction, is among the most crucial steps in predicting phase behavior of a hydrocarbon fluid system. Proper selection of single carbon number (SCN) distribution function is essential for heavy end characterization. The SCN distribution function is subject to fluid nature. The exponential distribution function has been and is widely applied to gas condensate plus fractions. More complicated functions are necessary in systems with jumps or discontinuities in successive SCN groups. Thirty fluid samples of a supergiant gas condensate reservoir are analyzed, most of which showing a discontinuity at SCN=10. A threecoefficient model is developed and then applied to determine the distribution function. The plus fraction is divided into three zones, each characterized by an adjustable parameter. A global optimization algorithm is developed and then applied to obtain unique coefficients for complete set of samples. This developed model predicts the experimental data with 10.6% accuracy and is in better agreement with experimental data compared to existing distribution functions.
Keywords
Main Subjects
1. Introduction
In simple mixtures, (e.g. systems that contain 23 components), the individual components are readily identifiable through the routine analysis techniques, while, many fluid systems in nature and in chemical industry are not readily identifiable because their mixture contains too many components, most of which with similar formula and close physical and chemical properties. This makes their analysis and separation difficult , yet impossible to full characterize and analyze for some mixtures (Danesh, 1998; K. S. Pedersen & Christensen, 2007). Petroleum fluids, polymer solutions, edible oil types, and coal tar liquids are examples of these complex mixtures. Compositional variations are gradual in complex mixtures, and composition of individual components may become hard to distinguish in some cases. Distribution functions are applied to describe compositional variations of successive components in these mixtures. Composition of petroleum fluids can be divided into two categories of the ones containing light components separately identify the ones known as "plus fraction", associated with continuous composition and characterized by distribution functions. For the second category distribution of compositions is described through equation (1), (Ahmed, 2007; Du & Mansoori, 1987; Mansoori, Du, & Antoniades, 1989):
(1) 
where, F(I) is the distribution function and I is an independent variable that can be either molecular weight or normal boiling point. The plus fraction and discrete components are weighted by and x_{i} mole fractions, respectively. The twoparameter exponential distribution function has been and is widely applied to characterize heavy end in gas condensate systems (Elsharkawy, 2003; K.S. Pedersen, Blilie, & Meisingset, 1992; K.S. Pedersen, Fredenslund, & Thomassen, 1989; Zuo & Zhang, 2000). In this function, the mole fraction of a component has a linear correlation with its molecular weight in a semilog coordinates system. In cases where jumps are evident in compositions of successive components, usually observed in SCN=8, the twoparameter exponential distribution function is no longer applicable. In these cases, the threeparameter gamma distribution function may describe the compositional distribution of plus fraction better (Whitson, 1983, 1984; Whitson, Anderson, & Soreide, 1990). This function increases calculation steps and requires parameters like molecular weight of plus fraction, which is reported as being uncertain in many cases. Ahmed et al. (Ahmed, Cady, & Story, 1985) proposed a simple model to describe the jump at SCN=8 by analyzing 34 oil and gas condensate fluid samples. They applied two linear functions for describing changes in molecular weight in SCN. The first line describes changes between SCN=78, and the second line considers the distribution from SCN=8 to the end. This approach avoids complexities associated with gamma distribution function. Hosein et al. revealed that the jump or discontinuity in SCN compositions is not limited to SCN=8 and may occur again in SCN=13 for some systems (Hosein, McCain, & Jagai, 2012; K.S. Pedersen, Thomassen, & Fredenslund, 1983, 1984). As the exponential and gamma distribution functions predict the normal distribution for compositions of SCN, they may underpredict the composition of SCN=13 and overpredict for SCN=12. To overcome this issue, Hosein et al. proposed a model to characterize the heavy end with fourcoefficient (4CM) by putting four linear functions in four divisions of heavy end compositional zones (Hosein et al., 2012). This approach is an extension of Ahmed et al (Ahmed et al., 1985) method which covers the jumps in both SCN=8 and 13. They applied the 4CM on 20 fluid samples separately and reported average values of model parameters calculated by minimizing the objective function for each one of the 20 samples. The 4CM proposed by Hosein et al. (Hosein et al., 2012) lacks a global optimization on selected samples, therefore, it is expected that applying the same parameters to other gas condensate fluid systems lead to considerable error.
Here, a global optimization procedure is proposed and applied on fluid samples taken from a supergiant gas condensate field to customize the model parameters for the field under study. In the following sections, the gas condensate field is described first, followed by a review of available pressurevolumetemperature (PVT) data. Next, methodology and model development for threecoefficient model (3CM) is developed and solution technique is described. Then, the results are presented and discussed. Concluding remarks appear at the last section of the article.
2. Available PVT Data
The gascondensate field under study is located at Persian Gulf, for which 87 PVT reports are collected between 19922013. A quality control protocol is applied on sampling conditions and PVT analysis of all samples, thirty PVT reports are found to pass the screening criteria reported by (Drohm, Trengove, & Goldthorpe, 1988; Moffatt & Williams, 1998). Tables 1a and 1b the properties of validated samples are tabulated in Tables 1a and 1b. The sampling depth and temperature were 27003450 (m) and 86110 (°C), respectively and the wells are distributed all over the whole reservoir area. Preliminary PVT studies reveal that reservoir fluid is lean gas condensate.
Table 1a. Composition of SCN Groups and Properties of C_{7+} Fractions for Samples 1 to 15 Applied on the Improved Model
SCN 
Well Name 

W1 
W2 
W3 
W4 
W5 
W6 
W7 
W8 
W9 
W10 
W11 
W12 
W13 
W14 
W15 

7 
0.520 
0.510 
0.480 
0.460 
0.470 
0.440 
0.480 
0.490 
0.450 
0.490 
0.480 
0.707 
0.677 
0.550 
0.503 

8 
0.510 
0.490 
0.450 
0.430 
0.440 
0.450 
0.420 
0.460 
0.370 
0.400 
0.410 
0.573 
0.560 
0.403 
0.364 

9 
0.340 
0.330 
0.290 
0.280 
0.290 
0.300 
0.290 
0.300 
0.220 
0.240 
0.254 
0.363 
0.346 
0.226 
0.206 

10 
0.260 
0.250 
0.220 
0.210 
0.220 
0.230 
0.210 
0.230 
0.190 
0.220 
0.252 
0.373 
0.351 
0.254 
0.221 

11 
0.180 
0.170 
0.150 
0.150 
0.150 
0.160 
0.150 
0.160 
0.150 
0.180 
0.177 
0.262 
0.250 
0.173 
0.152 

12 
0.130 
0.130 
0.117 
0.110 
0.106 
0.120 
0.110 
0.130 
0.120 
0.140 
0.135 
0.199 
0.186 
0.136 
0.119 

13 
0.120 
0.110 
0.100 
0.097 
0.094 
0.110 
0.100 
0.110 
0.100 
0.120 
0.128 
0.189 
0.177 
0.121 
0.107 

14 
0.090 
0.080 
0.075 
0.072 
0.069 
0.080 
0.070 
0.080 
0.070 
0.090 
0.096 
0.144 
0.136 
0.089 
0.079 

15 
0.070 
0.070 
0.063 
0.060 
0.057 
0.060 
0.060 
0.070 
0.050 
0.070 
0.076 
0.116 
0.109 
0.072 
0.066 

16 
0.050 
0.050 
0.045 
0.043 
0.041 
0.040 
0.040 
0.050 
0.040 
0.050 
0.053 
0.085 
0.080 
0.050 
0.047 

17 
0.040 
0.040 
0.037 
0.035 
0.033 
0.040 
0.030 
0.040 
0.030 
0.040 
0.045 
0.074 
0.070 
0.039 
0.039 

18 
0.040 
0.030 
0.031 
0.029 
0.028 
0.030 
0.030 
0.030 
0.020 
0.030 
0.038 
0.063 
0.060 
0.031 
0.032 

19 
0.030 
0.020 
0.025 
0.023 
0.022 
0.020 
0.020 
0.030 
0.020 
0.030 
0.029 
0.051 
0.048 
0.023 
0.026 

C_{20+} 
0.090 
0.070 
0.080 
0.070 
0.064 
0.060 
0.040 
0.070 
0.030 
0.070 
0.090 
0.180 
0.180 
0.040 
0.070 

Properties of Plus fraction: 

Z_{C7+} 
2.380 
2.280 
2.083 
2.070 
2.020 
2.130 
2.050 
2.250 
2.060 
2.390 
2.440 
3.620 
3.450 
2.410 
2.210 

MW_{C7+} 
139.126 
137.119 
138.479 
136.540 
134.970 
135.840 
132.860 
140.000 
136.000 
141.000 
139.580 
143.780 
143.980 
133.200 
136.870 

Table 1b. Composition of SCN Groups and Properties of C_{7+} Fractions for Samples 16 to 30 Applied on the Improved Model
SCN 
Well Name 

W16 
W17 
W18 
W19 
W20 
W21 
W22 
W23 
W24 
W25 
W26 
W27 
W28 
W29 
W30 

7 
0.480 
0.575 
0.575 
0.533 
0.533 
0.534 
0.575 
0.324 
0.401 
0.385 
0.454 
0.496 
0.552 
0.510 
0.565 

8 
0.353 
0.424 
0.424 
0.390 
0.390 
0.412 
0.406 
0.301 
0.310 
0.303 
0.371 
0.373 
0.382 
0.393 
0.383 

9 
0.193 
0.236 
0.236 
0.218 
0.218 
0.233 
0.234 
0.138 
0.165 
0.158 
0.191 
0.216 
0.219 
0.219 
0.223 

10 
0.216 
0.266 
0.266 
0.251 
0.251 
0.269 
0.265 
0.169 
0.186 
0.185 
0.219 
0.245 
0.246 
0.249 
0.254 

11 
0.149 
0.177 
0.177 
0.172 
0.172 
0.191 
0.178 
0.121 
0.128 
0.129 
0.153 
0.166 
0.166 
0.171 
0.176 

12 
0.119 
0.129 
0.129 
0.139 
0.139 
0.156 
0.141 
0.095 
0.101 
0.101 
0.120 
0.130 
0.133 
0.136 
0.144 

13 
0.100 
0.114 
0.114 
0.121 
0.121 
0.141 
0.122 
0.083 
0.087 
0.089 
0.105 
0.112 
0.118 
0.121 
0.131 

14 
0.073 
0.084 
0.084 
0.091 
0.091 
0.108 
0.092 
0.061 
0.064 
0.065 
0.078 
0.079 
0.088 
0.090 
0.098 

15 
0.062 
0.084 
0.084 
0.075 
0.075 
0.091 
0.076 
0.050 
0.051 
0.052 
0.063 
0.062 
0.073 
0.075 
0.083 

16 
0.042 
0.051 
0.051 
0.054 
0.054 
0.067 
0.055 
0.035 
0.036 
0.036 
0.045 
0.042 
0.053 
0.055 
0.061 

17 
0.033 
0.041 
0.041 
0.044 
0.044 
0.056 
0.045 
0.028 
0.028 
0.029 
0.036 
0.031 
0.043 
0.045 
0.050 

18 
0.028 
0.036 
0.036 
0.037 
0.037 
0.048 
0.038 
0.023 
0.023 
0.023 
0.030 
0.024 
0.036 
0.038 
0.043 

19 
0.021 
0.027 
0.027 
0.030 
0.030 
0.038 
0.030 
0.017 
0.017 
0.017 
0.023 
0.016 
0.029 
0.030 
0.034 

C_{20+} 
0.050 
0.090 
0.090 
0.090 
0.090 
0.120 
0.100 
0.047 
0.042 
0.047 
0.068 
0.030 
0.090 
0.090 
0.110 

Properties of Plus fraction: 

Z_{C7+} 
2.100 
2.540 
2.540 
2.430 
2.590 
2.650 
2.560 
1.640 
1.800 
1.770 
2.130 
2.200 
2.420 
2.400 
2.540 

MW_{C7+} 
134.220 
136.460 
136.460 
138.750 
140.640 
143.680 
138.600 
136.450 
133.710 
135.270 
137.010 
131.350 
137.930 
139.280 
141.230 

The composition of successive SCN for W29 and W16 samples. According to this figure and Tables 1a and 1b, the compositional jump, or discontinuity, for most samples is observed at SCN=10, rather than at SCN=8 and 13, as studied by Ahmed et al. (Ahmed et al., 1985) and Hosein et al. (Hosein et al., 2012). The average relative error of SCN predictions for all samples by these techniques are tabulated in Table 2, where, the former technique predictions show average relative errors of 7.0531.04%, while the latter shows average error within 4.4537.05% range. The large error of SCN prediction by these techniques and the discontinuity at SCN=10 necessitates the development of these models for the samples under study.
(b) 
(a) 
Figure 1. Molar Distribution of SCN Composition for Samples: (a) W29, (b) W16
Table 2. Average Relative error of SCN Predictions for All Samples by Ahmed et al. (Ahmed et al., 1985) and Hosein et al. (Hosein et al., 2012) Techniques
SCN 

C7 
25.69 
7.17 
C8 
19.44 
37.05 
C9 
31.04 
31.23 
C10 
10.00 
8.42 
C11 
7.05 
4.45 
C12 
10.44 
20.97 
C13 
23.46 
12.96 
C14 
20.02 
16.58 
C15 
23.84 
17.96 
C16 
14.69 
6.51 
C17 
17.76 
8.81 
C18 
22.93 
15.24 
C19 
23.64 
16.38 
Average 
19.23 
15.67 
3. Modeling
According to Ahmed et al. (Ahmed et al., 1985), composition of each SCN is calculated as a function of molecular weight and a composition of plus fraction as follows:
(2) 
The M_{n}, the molecular weight of SCN=n, is introduced by (Ahmed, 2007; Katz & Firoozabadi, 1978; Whitson, 1984). Normally, molecular weight and composition of C_{7+} fraction can be measured in the laboratory with accuracy. Here, Eq. (2) is applied in calculating the composition of heavier compounds by starting with C_{7+}. Based on the fluid nature and composition the molecular weight of C_{7+ }fraction is segmented into three different segments and a threecoefficient model (3CM) is developed for plus fraction, expressed as Eqs. (3)(5):
(3) 

(4) 

(5) 
where, M_{n+} and M_{n} are the molecular weights of plus fraction and n^{th} component at SCN=n, respectively. S_{1}, S_{2}, and S_{3} are the adjustable parameters calculated through global optimization of the overall objective function, OOF, defined by Eq. (6). In this approach, all samples are treated in a simultaneous manner.
(6) 
The objective function of an individual sample i , OF_{i}, is the sum of absolute relative errors in all SCN groups’ composition in this set of data, expressed through equation (7):
(7) 
The flowchart of calculations is shown in Fig. (2). A database of experimental data including molecular weight and composition of plus fraction, together with composition of SCN groups is constructed for all samples and is applied as the input of the model. An initial guess is made for model parameters, next, Eqs. (2 5) are solved for each sample to calculate composition of SCN groups and molecular weight of plus fraction. This procedure is run for C_{7+} to C_{20+} and SCN=7 to SCN=19. Composition of the last fraction is calculated through the material balance:
(8) 
Equation (7) is then applied to calculate the objective function for each sample. This procedure is repeated for all samples and global objective function is calculated using equation (6). The optimization algorithm, LevenbergMarquardt (Chandler, 1985), is then applied in minimizing the global objective function. The output of the proposed algorithm include optimum model parameters, composition of SCN groups and molecular weight of plus fraction for each SCN group, which are calculated for all samples in a simultaneous manner. Finally, the results of modeling are compared with the twoparameter exponential (K.S. Pedersen, Thomassen, & Fredenslund, 1989), Ahmed et al. (Ahmed et al., 1985), threeparameter gamma distribution function (Whitson, 1983) and 4CM (Hosein et al., 2012) models. Details of these models are given in Appendix.
Figure 2. Flow Chart of Global Optimization of Data based on Marching Technique (Hosein et al., 2012).
4. Results and Discussions
The statistical analysis of average SCN composition for all PVT samples are tabulated in Table 3, where, the standard deviations in average SCN composition is relatively high, indicating a considerable compositional variation in the same reservoir. The wide compositional variations among samples increase flexibility and wider range of model parameters.
Table 3. Statistical Analysis of Average SCN Composition for 30 Samples.
SCN 
Average composition 
Relative Standard Deviation (%) 
7 
0.507 
15.10 
8 
0.412 
15.56 
9 
0.246 
22.91 
10 
0.241 
17.62 
11 
0.168 
17.40 
12 
0.130 
17.05 
13 
0.115 
19.75 
14 
0.086 
21.54 
15 
0.071 
21.89 
16 
0.050 
23.09 
17 
0.041 
26.29 
18 
0.034 
28.75 
19 
0.027 
30.93 
C_{20+} 
0.079 
45.85 
The optimized parameters for the 3CM is given in table 4.
Table 4. Parameters of 3CM
SCN 

Parameter 
S_{1} 
S_{2} 
S_{3} 
Value 
15.7 
10.9 
13.3 
Applying the developed model, the molecular weight trend for plus fraction in the SCN=719 range is shown in Fig. (3) for one sample. This trend is divided in three regions of: I, showing the natural trend of molecular weight versus SCN for and IIIII, representing the molecular weight versus SCN for and by considering the discontinuities in SCN=9 and 10, respectively.
(a) 
(b) 
(c) 
Figure 3. Plus Fraction Molecular Weight Distribution. (a): Region I, , (b): Region II, , (c): Region III,
The average relative error of composition applying 3CM is calculated for all samples and the results are tabulated in table 5, along with the calculation results for twoparameter exponential (K.S. Pedersen, Thomassen, et al., 1989), Ahmed et al. (Ahmed et al., 1985), threeparameter gamma distribution function (Whitson, 1983) and 4CM (Hosein et al., 2012) models. The average absolute deviation (AAD%) is defined through Eq. (9):
(9) 
where, L is the total number of samples.
According to Table 5, the developed 3CM yields the minimum ARE% among all models. For the reservoir under study, the AAD% is 10.6% for 30 samples, while the twoparameter exponential (K.S. Pedersen, Thomassen, et al., 1989), Ahmed et al. (Ahmed et al., 1985), threeparameter gamma distribution function (Whitson, 1983) and 4CM (Hosein et al., 2012) models are 37.2%, 20.0%, 12.0% and 16.0%, respectively. The model validity is checked against composition of SCN groups and properties of C_{7+} fractions for two samples of Trinidad (Hosein & McCain, 2009) and North Sea (Ahmed et al., 1985). The absolute average deviations of SCN compositions for Trinidad and North sea samples are 16.2% and 13.7%, respectively.
The calculated results for the average composition of SCN groups by 3CM (column bars in white) are compared with average experimental SCN compositions (column bars in gray), Fig. (4), where, as observed the tolerances of the model prediction were also shown on each bar. Results indicatein this figure indicate an excellent match between experimental data and model predictions.
Table 5. AAD% of SCN Compositions with Different Models
Well Name 
Exponential (K.S. Pedersen, Thomassen, et al., 1989) 
Ahmed et al. (Ahmed et al., 1985) 
4CM (Hosein et al., 2012) 
Gama (Whitson, 1983) 
This work 
W1 
40.8 
14.0 
10.1 
8.4 
14.1 
W2 
34.6 
14.7 
11.0 
6.5 
10.1 
W3 
12.9 
12.7 
8.1 
8.6 
12.2 
W4 
11.5 
15.2 
12.3 
9.7 
11.4 
W5 
12.3 
15.6 
11.1 
13.6 
13.0 
W6 
16.9 
19.2 
14.6 
10.0 
12.4 
W7 
13.9 
23.0 
18.3 
11.4 
11.7 
W8 
23.3 
15.4 
10.8 
7.0 
10.3 
W9 
63.7 
24.2 
21.0 
14.2 
12.2 
W10 
15.4 
17.7 
15.8 
10.3 
8.2 
W11 
23.9 
19.5 
15.4 
11.0 
10.4 
W12 
84.8 
16.9 
13.0 
10.1 
9.0 
W13 
75.9 
15.5 
12.1 
9.9 
8.7 
W14 
34.5 
30.1 
25.6 
17.4 
12.4 
W15 
13.7 
20.8 
16.7 
13.7 
9.7 
W16 
32.9 
24.3 
20.0 
13.6 
11.3 
W17 
29.2 
18.4 
14.9 
14.1 
10.3 
W18 
22.0 
19.8 
15.7 
13.6 
9.9 
W19 
22.6 
20.8 
16.8 
12.7 
9.4 
W20 
22.6 
17.7 
14.5 
12.5 
7.0 
W21 
40.9 
20.8 
17.1 
11.8 
8.4 
W22 
30.2 
18.4 
14.7 
11.9 
8.6 
W23 
124.0 
24.2 
19.8 
13.6 
12.9 
W24 
100.6 
24.5 
20.1 
14.1 
12.0 
W25 
96.9 
23.5 
19.1 
12.9 
11.0 
W26 
22.4 
21.1 
16.8 
12.3 
9.8 
W27 
20.1 
31.0 
25.9 
16.6 
12.9 
W28 
20.6 
19.9 
16.2 
13.7 
9.7 
W29 
21.0 
20.9 
16.9 
12.6 
9.1 
W30 
31.5 
20.0 
17.0 
13.4 
9.1 
Average 
37.2 
20.0 
16.0 
12.0 
10.6 


Figure 4. Average Composition of SCN Groups. Solid Fill Column: Experimental Data, No Fill Column: Results of Developed Model




(a) 
(b) 




(c) 
(d) 


Figure 5. Distribution of Molecular Weight of Plus Fraction Versus SCN for Sample W29 with Different Models: Ahmad et al. (▬ ▪ ▬), 4 CM (▪▪▪▪), this work (▬▬), Experimental Data (▲). (a) Comparison of All Models, (b) Ahmad et al. Model (Ahmed et al., 1985), (c) 4CM (Hosein et al., 2012), (d) this work (3CM). 


The distribution of molecular weight of plus fraction versus SCN for sample W29 with different models is shown in Fig.(5). The results of Ahmed et al. (Ahmed et al., 1985) model, the 4CM (Hosein et al., 2012), and the developed model are compare with experimental data in this figure, where, the 3CM is in good agreement with experimental data. As observed, it is clear that the M_{n+} slope versus SCN changes in Ahmed et al. (Ahmed et al., 1985) to take into account the discontinuity of compositions at SCN=8, which is not the case for fluids of this reservoir. The 4CM (Hosein et al., 2012) and 3CM proposed here show three times and twice changes in original slope of molecular weight profile. Moreover, both Ahmed et al. (Ahmed et al., 1985) and 4CM (Hosein et al., 2012) models overpredict the experimental results, although the 4CM (Hosein et al., 2012) has three shifts in line slope to better match the data.
The effect of this overprediction on phase behavior of the sample is shown in Fig. (6), where, the PT results are calculated according to Ahmed et al. (Ahmed et al., 1985) and 3CM models for splitting plus fraction from SCN of 7 to 20.
It is important to note that the cross points, (i.e. boundaries between zones) coincide in this proposed 3CM, which is not the case in Ahmed et al. (Ahmed et al., 1985) and 4CM (Hosein et al., 2012) models. For example, in Ahmed et al. (Ahmed et al., 1985) model, the two linear functions proposed to describe the jump in SCN=8 do not yield the same values of M_{n+}. Similar results are obtained when the 4CM introduced by (Hosein et al., 2012) is applied. The discontinuity in predicting molecular weight of plus fraction at jump points is resolved in this proposed model by defining Eqs. (35) for different zones, as each zone applies the information of plus fraction nearest to it.
Figure 6. Effect of Distribution Function on PT Diagram for Sample W29. Ahmed et al. (▪▪▪▪) (Ahmed et al., 1985), and 3CM (▬▬) Models 
5. Conclusion
A global optimization is applied to a comprehensive set of gas condensate data set to characterize plus fractions that show discontinuity, or jump, in composition of SCN=10. It is revealed that the nature of plus fraction is important in selecting the proper distribution function. Therefore, sufficient number of PVT samples are necessary to determine the distribution function which yields the best characterization for heavy end. The proposed 3CM is tested on selected gas condensate samples and is revealed that this model can describe the SCN behavior of plus fraction better than its counterparts. The simplicity of this developed model together with global optimization where the data of all samples are applied, yield flexible, efficient and unique adjustable applicable parameters.
List of Symbols
A & B 
Constants in equation (A1) 
4CM 
Four Coefficient Model 
AAD% 
Absolute Average Deviation percent 
F 
Distribution function 
I 
Independent parameter of distribution function 
L 
Number of sample set 
M 
Molecular weight 
N 
Number of SCN groups 
OF 
Objective Function of each sample (well) 
OOF 
Overall Objective Function 
S 
Adjustable parameters of equations (3) to (5) 
SCN 
Single Carbon Number 
x 
Component mole fraction 
Z 
Molar composition of SCN group 


Greek Letter 

Distribution shape 

Variable in equation (A3) 

Minimum molecular weight of the plus fraction 



Superscripts 

Cal 
Calculation 
Exp 
Experimental 


Subscripts 

+ 
Plus fraction 
i 
Number of samples 
j 
Number of SCN groups 
n 
Number of SCN group 
Appendix: Distribution Functions
Exponential Distribution function:
Pedersen et al. proposed equation (A1) to correlate composition and molecular weight of SCN groups (K.S. Pedersen, Thomassen, et al., 1989):
(A1) 
and are the composition and molecular weight of SCN=i. Parameters A, B are the adjustable parameters, obtained by optimization methods. In this study A= 0.9813 and B= 0.0179.
Gamma Distribution function:
Whitson et al. proposed a threeparameter distribution function to describe the discontinuity in SCN=8, Eq. (A2) (Whitson, 1983, 1984; Whitson et al., 1990):
(A2) 
Γ is the gamma function, and α, M and are the distribution shape, molecular weight and minimum molecular weight of the plus fraction, respectively. is calculated through Eq. (A3):
(A3) 
where, M_{C7+ }is the molecular weight of C_{7+} plus fraction. By integrating (A2) the following is yield:
(A4) 
The mole fraction of SCN=I, , is calculated through Eq. (A5):
(A5) 
Ahmed et al. (Ahmed et al., 1985) distribution function:
Ahmed et al. (Ahmed et al., 1985) used equation (2) to calculate composition of SCN:
(2) 
To obtain discontinuity in SCN=8, two different distribution functions, with the same form as Eq. (A6) and different adjustable parameters, S are applied:
(A6) 
The adjustable parameter, S is presented in Table A1:
Table A1. Constant of Equation (A6) Proposed by Ahmed et al. (Ahmed et al., 1985)
SCN 
Gas condensate 
Oil 
n≤ 8 
15.5 
16.5 
n> 8 
17 
20.1 
The fourcoefficient model (Hosein et al., 2012):
In this model Eqs. (A7) and (A8) are applied to obtain the discontinuities in SCN=8, 13:
(A7) 

(A8) 
The plus fraction distribution function is divided into four zones, the constants of which are tabulated in Table (A2).
Table A2. Constant for equations (A7) and (A8) (Hosein et al., 2012)
SCN 
n=8 
8<n<13 
n=13 
n>13 
S 
12.5 
16 
13 
14.5 